FOM: cardinals in ZFC fragments

Kanovei kanovei at
Wed Aug 7 08:15:53 EDT 2002

> Date: Tue, 6 Aug 2002 15:59:01 -0400
> From: Harvey Friedman <friedman at>
> Scott showed that ZF handles cardinals as objects.
The "Hartogs cardinal" of a set X, equal to the set of all 
sets Y \in V_a equinumerous to X 
--- where a is the least ordinal such that the von Neumann 
set V_a contains a set equinumerous to X --- 
is a common formal definition of "the cardinal of X" in ZC. 
It requires the Foundation Axiom, but does not require AC. 

In ZF minus Foundation the Hartogs definition does not 
seem to work. 
I used to think that it had once established that 
ZF minus Foundation just does not admit any universal 
definition of "the cardinal of X", but in fact I know no 
any reference; 
no elementary proof is immediately visible either.


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